Theorem ttcwf2 37064

Description: If a transitive closure class is a set, then it is well-founded, assuming Regularity. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
ttcwf2	⊢ (TC+ 𝐴 ∈ V ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On))

Proof of Theorem ttcwf2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 487	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)))
2	1	eldifad 3916	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ∈ TC+ 𝐴)
3		ttctr2 37033	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ TC+ 𝐴 → 𝑥 ⊆ TC+ 𝐴)
4	2, 3	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ⊆ TC+ 𝐴)
5		dfss2 3922	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ TC+ 𝐴 ↔ (𝑥 ∩ TC+ 𝐴) = 𝑥)
6	4, 5	sylib 221	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → (𝑥 ∩ TC+ 𝐴) = 𝑥)
7		inssdif0 4328	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ TC+ 𝐴) ⊆ ∪ (𝑅1 “ On) ↔ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
8	7	bilanri 511	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → (𝑥 ∩ TC+ 𝐴) ⊆ ∪ (𝑅1 “ On))
9	6, 8	eqsstrrd 3971	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ⊆ ∪ (𝑅1 “ On))
10		vex 3458	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
11	10	r1elss 9776	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝑥 ⊆ ∪ (𝑅1 “ On))
12	9, 11	sylibr 237	. . . . . . . . 9 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ∈ ∪ (𝑅1 “ On))
13	1	eldifbd 3917	. . . . . . . . 9 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → ¬ 𝑥 ∈ ∪ (𝑅1 “ On))
14	12, 13	pm2.65da 828	. . . . . . . 8 ⊢ (𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) → ¬ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
15	14	nrex 3092	. . . . . . 7 ⊢ ¬ ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅
16	15	a1i 11	. . . . . 6 ⊢ (TC+ 𝐴 ∈ V → ¬ ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
17		difexg 5299	. . . . . . 7 ⊢ (TC+ 𝐴 ∈ V → (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∈ V)
18		zfreg 9556	. . . . . . 7 ⊢ (((TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∈ V ∧ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅) → ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
19	17, 18	sylan 591	. . . . . 6 ⊢ ((TC+ 𝐴 ∈ V ∧ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅) → ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
20	16, 19	mtand 827	. . . . 5 ⊢ (TC+ 𝐴 ∈ V → ¬ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅)
21		nne 2961	. . . . 5 ⊢ (¬ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅ ↔ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) = ∅)
22	20, 21	sylib 221	. . . 4 ⊢ (TC+ 𝐴 ∈ V → (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) = ∅)
23		ssdif0 4320	. . . 4 ⊢ (TC+ 𝐴 ⊆ ∪ (𝑅1 “ On) ↔ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) = ∅)
24	22, 23	sylibr 237	. . 3 ⊢ (TC+ 𝐴 ∈ V → TC+ 𝐴 ⊆ ∪ (𝑅1 “ On))
25		eleq1 2850	. . . . 5 ⊢ (𝑥 = TC+ 𝐴 → (𝑥 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On)))
26		sseq1 3961	. . . . 5 ⊢ (𝑥 = TC+ 𝐴 → (𝑥 ⊆ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ⊆ ∪ (𝑅1 “ On)))
27	25, 26	bibi12d 348	. . . 4 ⊢ (𝑥 = TC+ 𝐴 → ((𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝑥 ⊆ ∪ (𝑅1 “ On)) ↔ (TC+ 𝐴 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ⊆ ∪ (𝑅1 “ On))))
28	27, 11	vtoclg 3521	. . 3 ⊢ (TC+ 𝐴 ∈ V → (TC+ 𝐴 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ⊆ ∪ (𝑅1 “ On)))
29	24, 28	mpbird 260	. 2 ⊢ (TC+ 𝐴 ∈ V → TC+ 𝐴 ∈ ∪ (𝑅1 “ On))
30		elex 3475	. 2 ⊢ (TC+ 𝐴 ∈ ∪ (𝑅1 “ On) → TC+ 𝐴 ∈ V)
31	29, 30	impbii 212	1 ⊢ (TC+ 𝐴 ∈ V ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3088  Vcvv 3454   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  ∪ cuni 4871   “ cima 5663  Oncon0 6360  𝑅₁cr1 9732  TC+ cttc 37025

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-reg 9552

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-om 7861  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-r1 9734  df-ttc 37026

This theorem is used by:  ttcwf3  37065